Question

Prove that cos pi/16=1/2sqrtof 2+sqrt of 2+sqrt of 2?

Prove that `cos (pi/16)=1/2sqrt(2+sqrt(2+sqrt2))?`

Answer 1

`cos(pi/16)=1/2sqrt(2+sqrt(2+sqrt2))`

Explanation:

We know that,

`color(red)((1)cos(theta/2)=(1+costheta)/2, and cos(pi/4)=1/sqrt2)`

Also,...`pi/16=(pi/8)/2 and pi/8=(pi/4)/2`

Using (1) we get

`cos^2(pi/8)=(1+cos(pi/4))/2=(1+1/sqrt2)/2=(1+sqrt2/2)/2=(2+sqrt2)/4`

Taking squareroot of both sides,`( 0 < pi/8 < pi/2)`

`cos(pi/8)=(sqrt(2+sqrt2))/2`

Again using (1) we get

`cos^2(pi/16)=(1+cos(pi/8))/2=(1+(sqrt(2+sqrt2))/2)/2=(2+sqrt(2+sqrt2))/4`

Taking squareroot of both sides ,`( 0 < pi/16 < pi/2)`

`cos(pi/16)=(sqrt(2+sqrt(2+sqrt2)))/2=1/2(sqrt(2+sqrt(2+sqrt2)))`